Sadi Carnot stated that the efficiency of a reversible Carnot cycle is independent of the properties of the material used to run the cycle. Using this statement, all textbook discussions of the Carnot cycle use an ideal gas. Here, in contrast, we consider, in the spirit of the Caratheodory approach, a general analysis centered on the existence of an integrating factor that transforms an inexact differential into an exact differential. Also we derive a general relation between temperature and volume along an adiabatic path. Using these two results, we obtain the efficiency of the Carnot cycle, η = 1 − TC/TH, for any equation of state.